The present invention relates to fiber optic gyroscopes used for rotation sensing and, more particularly, to interferometric fiber optic gyroscopes.
Fiber optic gyroscopes are an attractive means with which to sense rotation. They can be made quite small and still be constructed to withstand considerable mechanical shock, temperature change, and other environmental extremes. In the absence of moving parts, they can be nearly maintenance free, and they have the potential to become economical in cost. They can also be sensitive to low rotation rates which can be a problem in other types of optical gyroscopes.
There are various forms of optical inertial rotation sensors which use the well-known Sagnac effect to detect rotation about a pertinent axis thereof. These include active optical gyroscopes which have the gain medium contained in an optical cavity therein, such as the ring laser gyroscope, and passive optical gyroscopes without any gain medium in the primary optical path, such as the interferometric fiber optic gyroscope and the ring resonator fiber optic gyroscope. The avoidance of having the active medium along the primary optical path in the gyroscope eliminates some problems which are encountered in active gyroscopes, such as low rotation rate lock-in, bias drift and some causes of scale factor variations.
Interferometric fiber optic gyroscopes typically employ a single spatial mode optical fiber of a substantial length, typically 100 to 2,000 meters, which length is formed into a coil by being wound on a core to form a closed optical path. An electromagnetic wave, or light wave, is introduced and split into a pair of such waves to propagate in opposite directions through the coil to both ultimately impinge on a photodetector. Rotation about the sensing axis of the core, or the coiled optical fiber, provides an effective optical path length increase in one rotational direction and an effective optical path length decrease in the opposite rotational direction for one member of this pair of electromagnetic waves. The opposite result occurs for the remaining member of the pair of electromagnetic waves for such rotation. Such path length differences between the pair of electromagnetic waves introduce a phase shift between those waves in interferometric fiber optic gyroscopes in either rotation direction, i.e. the well-known Sagnac effect. The use of a coiled optical fiber is desirable because the amount of phase difference shift due to rotation, and so the output signal, depends on the length of the entire optical path through the coil traversed by the two opposing directional electromagnetic waves. Thus, a relatively large phase shift difference can be obtained in a long optical fiber, but also in the relatively small volume taken by that fiber in its being coiled.
The output current from the photodetector system photodiode in response to the opposite direction traveling electromagnetic waves impinging thereon, after passing through the coiled optical fiber, follows a raised cosine function, that is, the output current depends on the cosine of the phase difference between these two waves. Since a cosine function is an even function, such an output function gives no indication as to the relative direction of the phase difference shift, and so no indication as to the direction of the rotation about the axis. In addition, the rate of change of a cosine function near zero phase value is very small, and so such an output function provides very low sensitivity for low rotation rates.
Because of these unsatisfactory characteristics, the phase difference between the two electromagnetic waves is usually modulated by placing an optical phase modulator on one side of the coiled optical fiber. As a result, one of the opposite direction propagating waves passes through the modulator just after entering the coil, while the other wave, traversing the coil in the opposite direction, passes through the modulator just before exiting the coil. In addition, a phase sensitive demodulator is provided to receive the photodetector output current. Both the optical phase modulator and the phase sensitive demodulator are typically operated by a sinusoidal signal generator, but other waveform types of a similar fundamental frequency can also be used.
The resulting signal output of the phase sensitive demodulator follows a sine function, i.e. the output signal depends on the sine of the phase difference between the two electromagnetic waves impinging on the photodiode, primarily the phase shift due to rotation about the axis of the coil. A sine function is an odd function having its maximum rate of change at zero, and so changes algebraic seen on either side of zero. Hence, the phase sensitive demodulator signal can provide both an indication of which direction a rotation is occurring about the axis of the coil, and the maximum rate of change of signal value as a function of rotation rate near a zero rotation rate. That is, the signal has its maximum sensitivity near zero phase shift so that its output signal is quite sensitive to low rotation rates. This is possible, of course, only if phase shifts due to other sources, that is, errors, are made sufficiently small. In addition, this output signal in these circumstances is very close to being linear at relatively low rotation rates. Such characteristics for the output signal of the phase 10 sensitive demodulator is a substantial improvement over the characteristics of the output current of the photodetector.
Reducing erroneous phase shifts from other sources is, however, a difficult problem in fiber optical gyroscopes. Avoidance of erroneous phase shifts in the electromagnetic waves reaching the photodetector requires that each of the interfering waves have traveled over the same optical path, that is, the electromagnetic wave associated with the clockwise direction of travel from the coil and the one associated with the counterclockwise direction of the coil each must travel over an indistinguishable optical path from the source to the photodetector absent any rotation of the coil. A system with this characteristic is often termed "reciprocal." At a minimum, the optical paths corresponding to the clockwise electromagnetic waves and the counterclockwise electromagnetic waves must be identical on an optical ray tracing basis in the absence of rotation. In meeting this requirement, a "minimum reciprocal configuration" has been found to be as shown in FIG. 1 in connection with the coiled optical fiber 10 shown there.
Coiled optical fiber 10 in FIG. 1 is, as indicated above, wound about a core or spool using a single mode optical fiber wrapped about an axis thereof which becomes the axis about which rotation is to be sensed. The use of a single mode fiber allows the paths of the electromagnetic waves to be defined nearly uniquely, and further allows the phase fronts of a such a guided wave to be defined uniquely. This greatly aids maintaining reciprocity.
In addition, the optical fiber is a so-called polarization-maintaining fiber in that a very significant birefringence is constructed in the fiber so that birefringence introduced by mechanical stress, which is unavoidable, and by the Faraday effect in magnetic fields, or from other sources, and which can lead to randomly varying phase difference shifts, become relatively insignificant. Thus, either the high refractive index axis, i.e. the slower propagation axis, or the low refractive index axis, i.e. the faster propagation axis, is chosen for primarily propagating the electromagnetic waves depending on the other optical components in the system. In the present system, the slow axis has been chosen in view of the optical components used therein, as will be subsequently described, and this slow axis will also be termed the "x" axis. The example given in FIG. 1 will be described with the slow axis always being the primary axis of electromagnetic wave propagation through the various optical fiber portions, but the opposite, fast axis, or "y" axis, could have been chosen, or there could be primary propagation over an optical path through such fiber portions which mixed the fast axis in some parts with the slow axis in other parts.
The coil is typically wound on a spool using the "quadrupole" technique so that similarly located points in the coil are near one another. This reduces the effects of time-varying phenomena, such as thermal gradients, from affecting opposite direction propagating electromagnetic waves differently from one another.
The electromagnetic waves which propagate in opposite directions through coil 10 are provided from an electromagnetic wave source, or light source, 11, in FIG. 1. This source is typically a superluminescent diode or, alternatively, a laser diode operating below its threshold for stimulated emission, either of which provide electromagnetic waves typically in the near-infrared part of the spectrum with a typical wavelength of 830 nm. Source 11 must have a short coherence length for emitted light to reduce the phase shift difference errors between these waves due to Rayleigh and Fresnel scattering at scattering sites in coil 10. Because of the nonlinear Kerr effect in coil 10, differing intensities in the two propagating waves can also lead to phase difference shifts therebetween. This situation can also be aided by the use of a short coherence length source for source 11 which leads to modal phase shift canceling. Rayleigh and Fresnel scattering, and the nonlinear Kerr effect, all lead to non-reciprocal phase shifts between the counter rotating electromagnetic waves in coil 10 even in a minimum reciprocal configuration. A superluminescent diode, or a laser diode operating below threshold, each have a wide emission spectrum compared to that of a laser diode operating past its threshold in the stimulated emission mode of operation.
Between laser diode 11 and fiber optic coil 10 there is shown an optical path arrangement in FIG. 1 formed by an extension of the ends of the optical fiber forming coil 10 to some optical coupling components which separate the overall optical path into several optical path portions. A portion of polarization maintaining optical fiber is positioned against a face of laser diode 11 at a location of optimum light emission therefrom, a point from which it extends to a first optical directional coupler, 12, to be joined thereto. If, on the other hand, coupler 12 is formed by fusing two optical fibers together in a coupling region, the excess length of one optical fiber may be positioned against diode 11 to provide this optical path between diode 11 and this wave coupling region of coupler 12, or the excess length may be spliced to another optical fiber extending from diode 11.
Optical directional coupler 12 has light transmission media therein which extend between four ports, two on each end of that media, and which are provided on each end of coupler 12 in FIG. 1. One of these ports has the optical fiber extending from laser diode 11 positioned thereagainst (or vice versa for a fused coupler). At the other port on the same end of optical coupler 12 there is shown a further polarization maintaining optical fiber positioned thereagainst (or alternatively extending from a fused coupler) which extends to be positioned against the photodiode, 13, which is electrically connected to a photodetection system, 14. In practice, as indicated above, coupler 12 may be formed from fused lengths of such optical fiber so that the remaining lengths past the fused portion, or the light coupling region therein, extend either all the way to laser diode 11 and photodiode 13 or are spliced to other optical fibers extending therefrom.
Photodiode 13 detects electromagnetic waves, or light waves, impinging thereon from the portion of the optical fiber positioned thereagainst (or extending thereto) and provides a photocurrent in response. This photocurrent, as indicated above, in the situation of two nearly coherent electromagnetic waves impinging thereon, follows a raised cosine function in providing 10 a photocurrent output which depends on the cosine of the phase difference between such a pair of electromagnetic waves. Photodiode 13 is operated in either the photovoltaic mode or the photoconductive mode, as needed, into an amplifier circuit of appropriate impedance to provide a photocurrent which is substantially a linear function of the impinging radiation intensity, and is typically a p-i-n photodiode.
Optical directional coupler 12 has another polarization maintaining optical fiber against a port at the other end thereof which extends to a polarizer, 15. Again, the excess length in an optical fiber past the coupling region in coupler 12 may have the end thereof extend all the way to polarizer 15, or may be spliced to another optical fiber portion extending from polarizer 15 with the principal birefringence axes in each such portion of optical fiber closely aligned to those of the other. At the other port on that same side of coupler 12 there is a non-reflective termination arrangement, 16, involving the excess length of one of the optical fibers fused together forming coupler 12 or, again, another optical fiber spliced to such an excess length.
Directional optical coupler 12, in receiving electromagnetic waves, or light, at any port, or at any end of an excess portion of optical fiber extending past the coupling region therein, transmits such electromagnetic waves so that a preselected fraction thereof, typically one-half, appears at each of the two ports, or ends of the two excess optical fiber lengths past the coupling region, which are at the opposite end of coupler 12 from that having the incoming port or excess optical fiber length receiving the incoming waves. On the other hand, no electromagnetic waves are transmitted to the port or excess fiber length which is on the same end of coupler 12 as is the incoming port. The polarization of the incoming electromagnetic waves with respect to the principle birefringent axes at the input port can be fairly well preserved at the corresponding axes of the two output ports, but there will be some coupling of waves between axes in the coupling region of the coupler.
Polarizer 15 is used because, even in a single spatial mode optical fiber, two polarization modes are possible for electromagnetic waves passing through such a fiber, one along each principle birefringent axis. Thus, polarizer 15 is provided for the purpose of transmitting the electromagnetic wave component along one of these axes, or one of these polarization modes, between the optical fibers connected to the ports on either end thereof, i.e. between the slow axes of those fibers as indicated above, while blocking transmission in the other mode between the fast axes of these fibers. Hence, the slow axis of the optical fiber extending from coupler 12 is aligned with the transmission axis of polarizer 15 at a port on one side thereof, or with the slow principal birefringent axis of an optical fiber connection portion extending from polarizer 15 that is closely aligned with the transmission axis of the polarizer. The fast axis of this optical fiber is then closely aligned to the blocking axis at the ports of the polarizer, or to the fast principal birefringent axis of a connection optical fiber extending therefrom which is closely aligned with the polarizer blocking axis.
Polarizer 15, however, is not capable of entirely blocking electromagnetic waves in the one state of polarization that it is intended to block. This shortcoming in the extinction coefficient thereof leads to a non-reciprocity between two opposite direction traveling waves over the optical paths they follow as will be further described below, and so to a small non-reciprocal phase shift occurring between them which can vary with the conditions of the environment in which the polarizer and the remainder of the system is placed. In this regard, the high birefringence in the optical fibers used in the system again aids in reducing this resulting phase difference as indicated above.
Positioned against the port of polarizer 15 on the end opposite that connected to optical directional coupler 12 is polarization-maintaining optical fiber portion which extends to a further optical directional coupler, 17, a coupler which has the same transmission properties as does coupler 12. Again, if directional coupler 17 is formed by fusing together two optical fibers, the excess portion of one of the optical fibers therein past the coupling region therein may extend all the way to the appropriate port on one end of polarizer 15 so that, typically, the slow axis of that fiber is aligned with the pass axis of polarizer 15, although the alternative of having the fast axis so aligned is also possible. In another alternative, this excess portion may be spliced to an optical fiber portion extending from polarizer 15 with the principal birefringent axes in each of the optical fibers joined in the splice being closely aligned to one another. Typically, again the fast axis on one side of such a splice will be closely aligned to the fast axis on the other side, but not necessarily so as the fast axis on one side of the splice could alternatively be aligned to the slow axis on the other side.
The second port on the same end of coupler 17 from which the first port is coupled to polarizer 15 is connected in a non-reflective termination arrangement, 18, using a further optical fiber portion or the excess length of optical fiber associated with that port beyond the coupling region in coupler 17. One of the ports on the opposite end of coupler 17 is connected to a further optical component in the optical path portion extending thereto from one end of the optical fiber in coil 10. The other port in coupler 17 is directly coupled to the remaining end of optical fiber 10 which is typically accomplished through a splice between the excess length of an optical fiber past the coupling region in coupler 17 and the optical fiber in coil 10, again with the principal axes of each closely aligned with those in the other, typically with fast axis to fast axis but possibly with fast axis to slow axis.
Between coil 10 and coupler 17, on the side of coil 10 opposite the directly connected side thereof, there is provided an optical phase modulator, 19. Optical phase modulator 19 has a port on either end of the transmission media contained therein which occur in FIG. 1 at the opposite ends of that phase modulator. The optical fiber from coil 10 is positioned against a port of modulator 19, with the principal birefringent axes of the optical fiber being closely aligned with the principal birefringent axes of that modulator. The optical fiber extending from coupler 17 is positioned against the port on the opposite end of modulator 19 with the principal birefringent axes thereon again closely aligned with those of that modulator, again fast to fast being typical but possibly fast to slow.
Optical phase modulator 19 can be of the variety formed by wrapping an optical fiber portion around a piezoelectric cylinder so that the fiber may be stretched by the application of voltage to that cylinder, or this phase modulator may be formed as an optical integrated chip using a substrate of lithium niobate, for instance, with metallic depositions provided thereon and adjacent a waveguide provided therein. Such depositions typically result in plate-like structures on the substrate to both provide electrical contacts to the modulator and a means through which varying electrical fields can be established in the waveguide to result in the necessary modulation of the phase of electromagnetic waves passing through that waveguide.
Optical phase modulator 19 is thus capable of receiving electrical signals on these plates to cause the introduction of phase differences in electromagnetic waves transmitted therethrough by changing the index of refraction of the transmission medium, or transmission media, because of the resulting electric fields established therein to thereby change the effective optical path length experienced by such waves. Optical phase modulators constructed in optical integrated circuit form have a large bandwidth, i.e. are able to provide phase changes following a waveform that has substantial high frequency content. Note also that polarizer 15 and source and loop optical directional couplers 12 and 17 could also be formed in similar integrated optic chips, including possibly being formed in a common such chip.
Directional optical coupler 17 serves as a beam-splitting apparatus in that electromagnetic waves emitted from source 11 that are transmitted through coupler 12 and polarizer 15 to be received by coupler 17 are there split in approximately half with a corresponding one of the resulting portions passing out of each of the two ports on the opposite end of coupler 17. Out of one port on that opposite end of coupler 17, the corresponding electromagnetic wave portion passes through optical fiber coil 10, through optical phase modulator 19, and back to coupler 17. A portion of that electromagnetic wave passes through the port of coupler 17 leading to polarizer 15 and then to coupler 12 where a part of the remainder of the wave portion is transmitted to photodiode 13.
The other portion of the electromagnetic wave after the split in coupler 17 leaves that other port on the coil 10 end of coupler 17 to first pass through optical phase modulator 19, and then optical fiber coil 10 to re-enter coupler 17 and, again, from there follow the same path as the first portion previously described to finally impinge in part on photodiode 13. In the presence of modulation provided by phase modulator 19, and in the presence of any rotation of coil 10 about its axis, or because of effects in coupler 17, some of the energy of the combined waves will be lost through non-reflective arrangement 18.
As indicated above, photodiode 13 provides an output current proportional to the intensity of the combined electromagnetic waves, or light waves, impinging thereon adjusted for the phase difference therebetween, and this photocurrent follows a raised cosine function in being based on the cosine of the average phase difference between each wave over the wavelengths present therein at its impingement on that diode. This relationship follows because the photocurrent depends on the resulting optical intensity of the two waves incident on photodiode 13 which intensity will vary depending on how much constructive or destructive interference occurs between the two waves at the diode. This interference of waves will change with rotation of the coiled optical fiber forming coil 10 about its axis as such rotation introduces a phase difference shift between the waves because of the Sagnac effect. Further, additional phase difference shifts can be introduced by optical phase modulator 19 as will be described in connection with the electrical system shown in the remainder of FIG. 1.
The electrical system portion of FIG. 1 shows an open loop fiber optical gyroscope system, but could also be converted to a closed loop fiber optic gyroscopic system, i.e. using feedback on the system. This would be accomplished by having the electrical system provide a feedback signal based on the output of the system shown in FIG. 1 to control a further optical phase modulator inserted in the optical path next to modulator 19, or to additionally control modulator 19. Optical phase modulator 19 is of the kind described above and is used in conjunction with a phase sensitive demodulator, or phase detector, for converting the output signal of photodiode 13 and photodetector system 14, following a cosine function, to a signal following a sine function. Following such a sine function provides, in that output signal, information both as to rate of rotation and as to direction of that rotation about the axis of coil 10. Modulator 19 is operated by a sinusoidal signal provided at the output of a bias modulation signal generator, 20, which also provides this signal to operate a phase detector which, as indicated, is a phase sensitive demodulator.
Thus, the output signal from photodetector system 14, including photodiode 13, is provided to an amplifier, 21, where it is amplified and passed through a filter, 22, to a phase detector, 23. The phase sensitive demodulator serving as phase detector 23 is a well-known device. Such a phase sensitive demodulator senses changes in the first harmonic, or fundamental frequency, of signal generator 20 to provide an indication of the relative phase of the pair of electromagnetic waves impinging on photodetector 13. This information is presented by phase detector 23 in an output signal following a sine function, i.e. the sine of the phase difference between the two electromagnetic wave portions impinging on photodiode 13.
Bias modulation signal generator 22, in modulating the electromagnetic wave portions in the optical path at a frequency set by the output signal supplied thereby as described above, also generates a strong second harmonic component in photodetector system 14. Filter 22 is a notch filter for removing this second harmonic component.
In operation, the phase difference changes in the two opposite direction propagating electromagnetic waves passing through coil 10 in the optical paths therethrough will lead to average net phase difference changes which will be relatively small, and which will vary relatively slowly compared to the phase difference changes due to optical phase modulator 19 and bias modulation signal generator 20. Any phase difference shift due to the Sagnac effect will merely shift the average phase difference between the electromagnetic waves, and the output signal from phase sensitive demodulator 23, after photodiode signal demodulation therein, will depending on the sine of this phase difference multiplied by an amplitude scaling factor set by the modulation of the waves due to phase modulator 19 and signal generator 20. This synchronous demodulation thus substantially extracts from the photodiode output signal the amplitude of the sinusoidal modulation frequency component at the modulation frequency introduced by signal generator 20 and modulator 19, which includes the results of any rotation of coil 10 about its axis, to provide the demodulator output signal.
As indicated above, however, additional phase shifts between the counter-propagating electromagnetic waves can be introduced even with the fiber optic gyroscope system in a minimum reciprocal configuration by various effects occurring therein. A significant source of such non-reciprocal phase shifts not due to the Sagnac effect is the following of differing optical paths by the two different polarization components of the counter-propagating electromagnetic waves. Such coupling of one polarization component of such a wave to the path followed by the other component occurs primarily at the coupling regions in couplers 12 and 17, at the splices described above between optical fiber optic portions, and at those interfaces where optical fibers are positioned against an optical path component such as source 11, couplers 12 and 17, and polarizer 15. Coupling of the polarization components of a wave occurs at these splice and interface locations because there is always some rotational misalignment of the principal birefringent axes of the optical fibers at a splice, or where they are positioned against an optical component such as the emitting face of source 11, even though such axes are mated at such splices and interfaces in close alignment.
Of course, the coupling of polarization components of such electromagnetic waves in one polarization optical path to the other would not be a problem if polarizer 15 were a perfect polarizing component permitting complete wave transmission in the transmission axis thereof and no transmission in the blocking axis thereof. However, polarizers, including polarizer 15 as indicated above, are not perfect and so are characterized by an extinction ratio, representing the fraction of an incident electromagnetic wave on the polarizer in the blocking axis which is found in that same axis at the output of the polarizer. Since, as described above, there are a substantial number of coupling points due to splices and the like, there are a substantial number of opportunities for the polarization components in an electromagnetic wave to take different paths, and further, to be coupled again back to the path originally followed. This situation, coupled with the imperfect polarizer, can lead to errors at photodiode 13 since such coupling results in effective non-reciprocal phase shifts.
Such errors in the transmission of electromagnetic waves in the system of FIG. 1 can be represented based on using a reference point between source coupler 12 and polarizer 15 in FIG. 1 since nothing fundamentally new or different will be added to these errors by including source coupler 12 and source 11. This reference point will be represented as a "wavy" line in FIG. 1 at which outgoing and returning electromagnetic waves will be compared. The electromagnetic waves at this "wavy" line reference point will be represented as E.sub.x (t) and E.sub.y (t) for the components traveling along the principal birefringent axes of the optical fiber extending from this reference point to polarizer 15. The "x" designation, as indicated above, indicates the slow axis electromagnetic wave component and the "y" designation indicates the fast axis electromagnetic wave component.
The electromagnetic waves returning to the reference point after having left the reference point to traverse through coil 10 and the optical components therebetween can be written as follows: EQU .sub.cw (.nu.)=G.sub.cw (.nu.)e.sup.+j.phi.r (.nu.) EQU .sub.ccw (.nu.)=G.sub.ccw (.nu.)e.sup.-j.phi.r (.nu.)
where .nu. is the optical frequency. The vector .sub.i (.nu.) (designated as such through having an arrow over the top thereof) represents the electromagnetic waves leaving the reference plane to travel through coil 10 and return, and so is composed of just the two scalars E.sub.x (t) and Ey(t) indicated above as occurring at the reference point but represented here by the Fourier transformation thereof, or ##EQU1## The transfer matrices representing the effects of the polarization maintaining fiber, polarizer 15 and coupler 17, operating on the representation of the departing waves .sub.i from the reference point, yields the returning wave vectors .sub.cw (.nu.) and .sub.cww (.nu.), can be written as follows: ##EQU2## The Sagnac phase shift is represented by 2.phi..sub.r.
As described above, the output signal provided by phase sensitive demodulator 23 depends on the total phase shift, .DELTA..phi., occurring between the counter-propagating electromagnetic wave passing through coil 10 to reach photodetector 13. Thus, this output will depend essentially on the phase difference of the waves returning to the "wavy" line reference point which can be found from the argument of the complex matrix resulting from the product of the two waves, or EQU .DELTA..phi.(.nu.)=arg[ .sub.ccw (.nu.) .sub.cw (.nu.)]
The "dagger" symbol indicates that the Hermitian conjugate of the matrix is being used. This last equation can be rewritten using the equations above for .sub.cw (.nu.) and .sub.ccw (.nu.) and taking .phi..sub.r to be zero, i.e. no rotation of coil 10 about its axis perpendicular to the plane of the coil in FIG. 1, so only phase differences due to errors remain. Then, the phase differences, or path difference phase errors, .DELTA..phi..sub.e, due to polarization component path differences appear as: EQU .DELTA..phi..sub.e (.nu.)=arg[ .sub.i (.nu.)G.sup. .sub.ccw G.sub.cw .sub.i (.nu.)]
Here again, the indicates that the Hermitian conjugate of the matrix is being used. This last result is obtained at each optical frequency v, and so is the end result if a near monochromatic optical source is used for source 11. However, since source 11 is typically going to be a "broadband" optical source, the total error can be found only through integrating the last equation over the optical frequency although such a step is not made explicit here.
The off-diagonal elements of the transfer matrices [G.sub.cw ] and [G.sub.ccw ] in the system of FIG. 1 are much smaller in magnitude than the diagonal terms therein in a polarization maintaining optical fiber based gyroscope using a polarizer. Since these off diagonal terms in these matrices are small, the last equation can be shown to be approximately: EQU .DELTA..phi..sub.e .apprxeq..DELTA..phi..sub.ampl +.DELTA..phi..sub.inten
where: ##EQU3## That is, the total phase error .DELTA..phi..sub.e at any optical frequency .nu. can be separated into two parts, an amplitude related phase error, .DELTA..phi..sub.Ampl, and an intensity related phase error, .DELTA..phi..sub.Inten. The last two expressions show that amplitude related phase errors depend on the relative phases of the electromagnetic wave polarization components .sub.k (.nu.) and .sub.y (.nu.) and the first power of the polarization extinction coefficient, whereas the intensity related phase error depends on the differences in optical power in the two polarization component optical paths and the square of the polarizer extinction coefficient.
Elimination or significant reduction of such errors has been attempted in several ways. Relatively long sections of polarization maintaining optical fiber have been inserted in the optical path for this purpose, or a long section has been inserted followed by a polarization component intensity equalizer between source coupler 12 and polarizer 15. Birefrigence modulators have been inserted in the optical path.
Such measures can be expensive both in component costs and in assembly costs. Thus, there is a desire for a measure or measures to reduce or eliminate such errors at more moderate cost.